Advanced Topics in Equilibrium Statistical Mechanics

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1 Advanced Topcs n Equlbrum Statstcal Mechancs Glenn Fredrckson 10. Electrolyte and Collodal Solutons A. Systems and Models Thus far, we have restrcted our attenton to the equlbrum SM of smple fluds and polymers. However, there are many other systems of general nterest to chemcal engneers: collodal solutons: 1-1 electrolyte soluton: polyelectrolyte soluton: We model these systems a bt dfferently, dependng on the lengthscales nvolved. In many cases, t s convenent and accurate to replace the solvent by a contnuum, so that we do not have the added computatonal expense of descrbng the solvent structurng around the solute. Let s start wth the smplest case of a soluton of neutral partcles. Imagne that we have a soluton of non-charged, neutral molecules or collods, dssolved n a low molecular weght solvent: 1

2 If such systems are dlute n solute, then we mght thnk about usng a vral expanson to address the thermodynamc propertes. The effectve nteractons between solute molecules would correspond to the potental of mean force w(r) between two solute molecules separated by some dstance r n the pure solvent. Indeed there s a theory known as the McMllan-Mayer theory that makes a rgorous connecton between dlute gases and dlute nonelectrolyte solutons: Gases p k B T = ρ + B n (T )ρ n EOS n= u(r) par potental n vacuum Dlute Non-electrolyte solutons Π k B T = ρ + Bn(T,µ)ρ n n= EOS w(r j µ): potental of mean force between two solutes at separaton r n pure solvent at chemcal potental µ. ρ: solute concentraton # densty) Π: osmotc pressure of the soluton The Bn are related to w n precsely the same way that B n are related to u, e.g B(T,µ)= 1 dr[e βw(r,µ) 1] Thus, dlute non-electrolyte solutons are no more dffcult to deal wth than gases, provded we can make good models of w(r; µ). For, large ( 1 µm) collodal partcles, t s sensble to neglect the granularty of the solvent and model w by a smple hard sphere potental. For stercally stablzed collods wth grafted polymers, we mght want to employ a softer repulsve potental wth a longer range startng at R. Fnally, for molecular solutes, one mght do a MC or MD smulaton of solutes n the solvent to explore w(r; µ). Next let s consder a smple electrolyte soluton, e.g. NaCl n water a socalled 1-1 electrolyte. The crudest descrpton of such a soluton s the so-called prmtve model:

3 Treat solvent as a contnuum wth delectrc constant ɛ. Let 1 be the caton (+) speces and be the anon( ) speces. Model ther nteractos as the sum of a hard sphere and coulomb potental: e.g., {, r < 1 w 1 (r, µ) = (σ 1 + σ ) q 1,q ɛr, r > 1 (σ 1 + σ ) Here, σ 1 and σ are the caton and anon hard sphere dameters; q 1 and q are the charges (q 1 =+e, q = e for NaCl). Ths model clearly makes a drastc approxmaton for the short-ranged part of w j (r, µ). However, we shall work wth t, snce we shall see that the long-ranged coulomb nteracton s actually domnant at low concentratons. Let s now consder the so-called restrcted prmtve model where σ 1 = σ = σ for smplcty. One mght thnk that the McMllan-Mayer theory should apply, so we could develop a vral expanson: Π k B T = ρ 1 + ρ + =1 j=1 Bj(T,µ)ρ ρ j +... Bj(T,µ)= 1 dr[e βwj(rjµ) 1] where Bj s a matrx of second vral coeffcents. The problem wth ths s that these vral coeffcents dverge for the coulomb potental: Bj nt σ dr r [ e βq q j ɛr 1 ] 1 r for r r for r = Clearly vral expansons fal for electrolyte solutons! A new approach s needed. We shall see that the problem s that the osmotc pressure s not analytc n the ρ for charged solutons. Instead, Π k B T ρ 1 + ρ + θ(ρ 3/ ) The Debye-Hückel theory was developed to evaluate ths θ(ρ 3/ ) electrostatc term. B. Debye-Hückel Theory There are two ways to deal wth the dvergences caused by electrostatc nteractons: 3

4 Mean-feld approxmatons descrbed n Chapter 15 of McQuarre Summng nfnte classes of dvergent dagrams see the Andersen handout. We shall follow the second approach here, although both lead to the same lmtng correctons to deal gas behavor for dlute electrolytes. Recall from the Andersen artcle that βa ex β[a(t,v,n 1,N ) A d (T,V,N 1,N )] has the smple dagrammatc seres: sum of top dstnct, rreducble βa ex graphs wth two or more ρ-feld = ponts, and at most one f-bond between each par of ponts. = Note that rreducble connected + no artculaton vertces Lecture 19 Debye-Hückel Theory for electrolyte soluton: βa ex = where f j (r) =e βwj(r) 1, speces 1, e.g., Na + speces, e.g., Cl W j (r) =Wj d qqj (r)+ ɛr e γr θ(r σ) { = 0 r<σ 1 r>σ hard sphere d = σ convergence factor γ 0+ 4

5 In ths expresson, each f j bond represents: f j (r) =e βwj(r) 1 and we sum over all speces ndces when we evaluate dagrams. For example: Let s now wrte W j (r) = = 1 dr 1 dr f j (r 1 )ρ ρ j =1 j=1 Wj(r) d + hard sphere d = σ By expandng the Coulomb potental: q q j ɛr e γr coulomb wth a convergence factor, γ 0+ f j = fj d +[1+f j d ] (ϕ j ) n /n!, n=1 ( ) bond fj d d e βw j 1 f d ϕ j β qqj ɛr e γr θ(r σ)( ) bond = θ(r σ) { 1, r > σ 0, r < σ It can be argued that ths substtuton n the dagrams leads to (see Andersen): sum of top. dstnct, rreducble graphs that have or more ρ-feld βa ex = ponts, and at most one f d -bond and any number of ϕ-bonds between each par of ponts = βa ex d + where A ex d contans all the dagrams wth only dashed f d bonds. Notce that f lϕbonds connect any two ponts, there s now a factor of 1 l n the value of a dagram! Let s now look at some ndvdual dagrams: βa ex d + θ(vρ 3 ) Vσ 3 ρ domnant term for η σ 3 ρ 1 dlute 5

6 But, = 1 dr 1 dr ϕ j (r 1 )ρ ρ j j = 1 βv ( q ρ ) r>σ dr 1 ɛr e γr q ρ = ɛρ 1 eρ = 0 whch s a statement of electroneutralty. Thus =1 =0. Now, = 1! β V ( ) q ρ dr 1 r>σ ɛ r e γr γ 1 for γ 0 ( ) = 1 βv q ρ drf d (r) 1 ɛr e γr θ(r σ) 0 0 ( ) = 1 3! β3 V q 3 ρ = 1 6 β3 V } {{ } 0for 1 1 electrolyte ( r>σ dr 1 ɛ 3 r 3 e 3γr } {{ } ln(γσ),γ 0 q ρ ) 3 dr1 dr3 1 ɛ 3 1 r 1r 13r 3 e γ(r1+r13+r3) θ(r 1 σ)θ(r 13 σ)θ(r 3 σ) γ 3 for γ 0 By followng ths path, t can be shown that the largest dagrams for γ 0at each order n ρ are the rng dagrams : βa ex rng = We can evaluate ths seres by usng Fourer transforms; forγ 0, we can gnore the θ-functons. Notce these dagrams all look lke: 6

7 So let s group thngs ths way, defnng ( ) κ 4πβ q ρ ; ψ(r) 1 ɛ 4πr e γr Then or βa ex rng = V (κ ) { dr 1 ψ(r 1 1 ψ(r 1) +( 1) κ dr 3 ψ(r 13 )ψ(r 3 ) +( 1) 4 (κ ) 4 dr3 dr4 ψ(r 13 )ψ(r 34 )ψ(r 4 )+... } By the convoluton theorem: βa ex rng = V (κ ) dr 1 ψ(r 1 )C(r 1 ) = V (κ ) 1 3 π dk ˆψ( k) Ĉ(k) ˆψ(k)= dre k r 1 4πr e γr = 1 γ +k Ĉ(k)= 1 1)3 ˆψ(k)+ 3 κ [ ˆψ(k)] + ( 1)4 4 (κ ) [ ˆψ(k)] = 1 ˆψ { ( 1) + ( 1)3 3 κ ˆψ + ( 1)4 4 (κ ) ˆψ } +... { } = 1 ˆψ ( 1) n (κ ˆψ) n (κ n ˆψ) n= κ ˆψ+ln(1 κ ˆψ) So βa ex rng = V (π) 3 Let q = k/κ, γ 0+ dk{κ ˆψ(k) + ln(1 κ ˆψ(k))} A few comments: βaex rng V = κ3 (π) dq{q +ln(1 q } 3 = κ3 1π ρ3/ 1. The sume of the rng dagrams s fnte for γ 0 (coulomb potental), even though the ndvdual terms are dvergent. Mayer (see Andersen) showed that the entre A ex seres can be renormalzed, effectvely replacng ϕ bonds by C bonds and elmnatng all long-ranged dvergences.. All remanng dagrams n the renormalzaton seres, as well as A ex d /V, are θ(ρ ) whch are smaller than the rng sum θ(ρ 3/ ). Thus the rng sum gves the frst correcton to IG behavor for ρ 0. Note that the fnte sze of the ons, σ, does not enter untl θ(ρ ). 7

8 In summary, we have derved the followng Debye-Hückel expresson for A V = A/V : βa V = ρ ln ρ + ρ (ln λ 3 T, 1) κ 1π deal gas +θ(ρ ) where κ = 8πβ ɛ I,I 1 q ρ onc strength and ξ D κ 1 I 1/ Debye screenng length We shall dscuss the physcal meanng of ξ 0 = κ 1 shortly, but frst we note that ) βπ= β A V T,N = βa V + βa V ρ ρ = ρ κ3 4π + θ(ρ ) as prevously announced! The chemcal potentals are: βµ β A ) =ln(ρ λ 3 N T,) β κq ɛ V,T Recallng the defnton of the actvty coeffcent γ : µ = k β T ln(λ 3 T,ρ γ ) we fnd a purely electrostatc contrbuton: ln γ = κq ɛk β T I As shown n the attached fgures, ths lmtng law works ncely at low concentratons, but typcally breaks down around ρ 0.01 moles/l. Our next task s to explan the sgnfcance of the Debye length ξ D = κ 1. We shall do ths by studyng the structure of an electrolyte soluton,.e. g j (r). Frst, however, note that the electrostatc contrbuton to both A V and Π can be wrtten: A el V Π el k B Tκ 3 k BT ξ 3 D Ths s remnscent of our blob pcture of crtcal phenomena and sem-dlute polymer solutons! 8

9 Our startng pont for examnng the structure of a dlute electrolyte soluton s the OZ equaton: h j (r 1 )=c j (r 1 )+ k=1ρ k dr 3 c k (r 13 )h kj (r 3 ) In the dlute lmt of ρ 0, c j (r) f j (r) =e βw 0 j (r) 1 where Wj 0 (r) s the potental of mean force between two ons n the pure solvent. Moreover, snce ons are far apart n the dlute lmt: c j (r) βwj(r) 0 = β q q j ɛr Smlarly, we can wrte h j (r) g j (r) 1 e βwj(r) 1 βw j (r) where W j (r) s the potental of mean force between two ons when surrounded by other ons at concentraton {ρ,ρ }. Wth these approxmatons, the OZ equaton becomes W j (r 1 )= q q j q q k β ρ k dr 3 W kj (r 3 ) ɛr 1 ɛr 13 Defne W j (r) qqj ɛ W (r) W (r 1 )= 1 κ 1 dr 3 W (r 3 ) r 1 4π r 13 Ths s a lnear ntegral equato that can be solved by Fourer transforms: k=1 Ŵ (k)= 4π k κ 1 k Ŵ (k)= 4π k +κ Ŵ (k) W (r)= 1 (π) dke k r Ŵ (k) = e κr 3 r W j (r) = qqj ɛr e r/ξd Thus, we see that whle the bare coulomb potental s long-ranged: W 0 (r) = 1 r the potental of mean force between two ons s screened by the surroundng ons: W (r) = 1 r e r/ξd On lengthscales greater than ξ D = κ 1 I 1/, two ons no longer feel each other! The physcal pcture s: 9

10 Notce that ξ D 30 Åatρ =0.01 mole/l, whch explans why σ 3 Ås not relevant. Also note that the thermodynamcs s evdently also controlled by these clouds, whch play the role of blobs, Π el A el V k BT/ξ 3 D! Lecture 0 Recap: Debye-Hückel Theory For dlute electrolyte solutons: βa ex V = κ3 1π + θ(ρ ) κ = 8πβ ɛ I,I 1 q ρ onc strength ξ D κ 1 I 1/ Debye length Wj 0 = qqj ɛr bare coulomb nter. W j (r) = qqj ɛr potental of mean force e r/ξd C. Applcaton to Charged Collods A common strategy for preventng the aggregato nof collodal partcles n soluton (due to attractve VDW nteractosn) s to mmoblze charges on ther surface. These charges are almost always derved from a neutral speces that dssocates, so counter ons are also present: Ths stuaton can mmedately be treated by usng Debye-Hückel theory, provded: 1. The system s dlute n solute (collod),.e., η s 1. The surface densty of charge on the collod s not too hgh so that the local charge densty n soluton does not nvaldate DH theory. 10

11 If these condtons are met, and the total charge on the collod s +Q and the counter ons are monovalent, then we can vew the soluton as a Q 1 electrolyte: Provded ξ D σ, we can smply model the effectve nteractons between two collods as: w(r) = W S (r) + Q ɛr e r/ξd short-ranged potental long-ranged [hard-sphere (da. σ) electrostatc and attractve VDW] repulson We could then use W (r) as an effectve par potental and carry out MD or MC smulatosn to determne structure and thermo props of the soluton. In such a smulaton, only the collod postons are retaned the water molecules and counter ons enter only as a contnuum through W (r)! Notce that the total onc strength enters ξ D : [ ξ D = κ = 4πβ ɛ q ρ ] Q ρ + e Q e ρ (Q + eq)ρ Q 1 electrolyte No added salt where ρ s the collod number densty. It s common to add an extra free salt, e.g. NaCl to ncrease the onc strength and thereby decrease ξ D the range of W (r). Then: ξ D I (Q + eq)ρ + S q Sρ S 11